FCSXpert Solutions: Fluorescence Correlation Spectroscopy Simplified!.
## FCS Classroom

#### Autocorrelation of Fluorescence Intensity

### Detailed Derivations

#### Relationships Between Fluctuations and Averagesback to top

#### Correlations Between Fluctuations and Between Averagesback to top

#### Correlations Between Fluctuations and Between Averages in Timeback to top

In this tutorial we will derive the equations that govern autocorrelation of fluorescence intensities. The important points of this section are:

- For random processes such as diffusion, the average of the square of the fluctuations in a parameter equals the average parameter value, that is
- As a result, if one has a physical parameter such as concentration, C, or intensity, I, which is directly proportional to N,
- We define the intensity correlation function, g(Δt), to be the correlation coefficient between the fluctuation at time τ from the mean intensity and the fluctuation at some time interval, Δt, later, over all values of measurement time,τ:
- Similarly, we define the intensity correlation function, G(Δt), to be the correlation coefficient of the intensity at time τ and the intensity at some time interval, Δt, later, over all values of measurement time,τ:
- The values of g(Δt) and G(Δt) at Δt = 0 are then given by

See the Detailed Derivations below for in-depth mathematical derivations of these relationships.

Let us begin by considering an important property of random processes, such as translational diffusion.
The concentration of a beaker containing a number of molecules is a constant
because the total number of molecules in the beaker is constant.
However, if we focus on a very small subvolume, such as the confocal volume, V_{C}, often used in fluorescence correlation spectroscopy (FCS)
(see What is the Confocal Volume?),
then there will be an average number of molecules in the volume, <N>, and fluctuations, say due to random diffusion,
of this number about this average which we shall call δN. It is a property of randomly distributed processes that

, (1)

where <(δN)^{2}> is the sample variance calculated as follows:

. (2)

If one has a property like concentration or fluorescence intensity that is proportional to the number of molecules, then

, (3)

where <I> is the average fluorescence intensity and q is a multiplier.

If we “normalize” by dividing by <I>^{2} = q^{2}<N>^{2}, we get that

. (4)

This is a convenient way to eliminate the proportionality constant from the relationship between intensity fluctuations and average particle number by incorporating the easily calculated average intensity quantity, <I>.

Equation 2, above, is reminiscent of the definition of the correlation coefficient, R, between two variables x and y:

. (5)

If x = y, then the variables are completely correlated and we get

. (6)

In Equation 4, above, we introduced a slightly different normalization that eliminates the proportionality constant. If we use this normalization to define a new coefficient, g, we get

, (7)

where our normalization incorporates two variables, x and y, instead of only one variable, I (as in Equation 4).

If x = y, g does not go to 1, but instead reduces to

. (8)

If we make x = y = I, Equation 8 becomes Equation 4, and we get

. (9)

As an alternative measurement of correlation, instead of calculating g (Equation 5), the relationship between the fluctuations δx and δy, we can calculate, G, the relationship between x and y, such that

. (10)

Now, if we make x = y = I, Equation 10 becomes

. (11)

We know that each intensity value can be defined in terms of its deviation from the average intensity as I = <I> + δI. Plugging this relationship into Equation 11 yields

. (12)

Recognizing that because random processes exhibit as many deviations below the mean as above it (<δI> = 0), we have that

. (13)

Finally, using the relationship from Equation 4 again, we get

. (14)

We can generalize this a bit and consider not just the correlation of intensities with themselves, but between intensities and the intensities at some time, Δt, later. We define these so called autocorrelation functions as

. (15)

Where the relationship between G(Δt) and g(Δt) still holds:

. (16)

For the case that Δt = 0, we are back to Equation 14, such that

. (17)